The $S_\bullet$-construction as an equivalence between 2-Segal spaces and stable augmented double Segal spaces

TL;DR

This paper explores a generalized S-construction that establishes an equivalence between 2-Segal spaces and stable augmented double Segal spaces, advancing the understanding of their structural relationship in higher category theory.

Contribution

It provides a streamlined exposition of a result linking 2-Segal spaces with stable augmented double Segal spaces via a generalized Waldhausen S-construction.

Findings

Establishes an equivalence between 2-Segal spaces and stable augmented double Segal spaces.

Clarifies the role of the generalized S-construction in higher categorical structures.

Connects concepts relevant to algebraic K-theory and Hall algebras.

Abstract

This note is a contribution for a proceedings volume of the workshop "Higher Segal Spaces and their Applications to Algebraic K-Theory, Hall Algebras, and Combinatorics". The content is a streamlined exposition based on a talk about a result by Bergner-Osorno-Ozornova-Rovelli-Scheimbauer from WITII. We discuss how a generalized version of Waldhausen's S-construction describes a correspondence between 2-Segal spaces and certain double Segal spaces, which satisfy further conditions of stability and augmentation.